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Introduction

Non-asymptotic numerical simulation of transient heat transfer requires knowledge of initial conditions. If the heat transfer medium is soil, they typically use temperature logs (depth-temperature tables) of monitoring wells for the initial time point. Then one may either use scattered-data interpolation [1] or solve a steady heat transfer problem with Dirichlet boundary conditions at temperature measurement points. If you choose the second way and the problem is nonlinear, there might be some convergence issues.

For the mathematical simulation of thermal field distribution in the ground, it is necessary to account for convective heat transfer (heat transfer by means of mass transfer). Convective heat transfer is caused by the filtration of water into the ground resulting, for example, from precipitation. The temperature distribution is described by the partial differential equation of heat conduction where, in the case of convection, there is a so-called convective term. Since the computational domain is arbitrary, the heat equation is solved numerically by using finite-difference methods (FDM). The Douglas – Rachford ADI scheme is one of these methods. In this paper, we focus on modification of the scheme to account for convection in the computational domain.

It is well known that the foundations of buildings are subject to loads which can result in deformation and subsidence. Hence, the analysis of foundation deformation must be conducted at the design stage. This article describes the computer simulation of foundation deformation. We propose an approach based on numerical solution of the stationary differential equation in partial derivatives. This equation describes the transversal deflection of a thin plate (foundation slab), taking elasticity into account, due to an external orthogonal force.

2. Plate Deflection Equation

Let the Cartesian coordinate system be the plate plane.By we define the domain of the plate in this plane. Let be the boundary of the domain. The function for plate deflection is given by . At small transversal (vertical) deflections, the function satisfies the following equation [1]:

This article describes the performance of calculations on video cards (using CUDA) for modeling physical processes and phenomena based on the solution of the three-dimensional heat equation via the Douglas-Rachford scheme (ADI method). A comparative analysis of the calculation speeds of the central (CPU) and graphics (GPU) processors was conducted.

2. DOUGLAS-RACHFORD SCHEME DESCRIPTION

For the mathematical simulation of heat distribution, accounting for filtration and phase transition, the following heat equation is applied:

The article is devoted to methods of base settlement on permafrost calculation with detailed description according to SNIP 2.02.04-88.

INTRODUCTION

The calculation of bases in permafrost regions is quite a complicated and specialized process, heavily influenced by thermal field and thawing phenomena. Thermal field causes permafrost thawing and decreases its load bearing capacity while increasing soil base deformation, which is usually evidenced in the form of base settlement. In fact, to determine the deformations during building maintenance, it is necessary to solve for the stress-strain state of the base, which is described by differential equations of equilibrium and the laws of elastic-plastic deformation.

Many software packages for numerical computations allow users to use a static adaptive (hereby referred to as adaptive) step in the construction of an orthogonal hexahedral structured computational mesh. This means that informed users can employ their experience to get a more accurate computation without significantly increasing the computation time by specifying the areas of the computational domain in which, in their opinion, it is necessary to apply more detailed partitioning (use a smaller spatial step) as compared to the rest of the computational domain.

When properly used, adaptive partitioning of the computational domain is a powerful tool in numerical computations to increase accuracy. However, when the above option is overused, the computational time can increase dramatically without necessarily altering the accuracy of the computation to any significant degree. In this article, we describe the theoretical advantages and disadvantages of using adaptive partitioning of the computational domain, and also give two examples for numerical computations of thermal fields in soils. In the first example, application of the adaptive step is appropriate; this is not the case, however, in the second.

Abstract — The quasilinear heat equation with thermal conductivity and heat capacity depending on the temperature field in three spatial dimensions is studied in application to the phase transition problem in permafrost soils. The conditions under which the alternating directions Douglas – Rachford finite difference scheme retains numerical stability are explicitly formulated. The comparison with the known analytical similarity solution to the Stefan problem in one spatial dimension is performed.

Since the early formulations of the alternating directions implicit methods (ADI methods) [1], [2], they have been tremendously developed and found a vast number of applications [3], [4]. Nevertheless, serious difficulties are encountered with the use of these methods in application to problems with complex geometries [5] and/or nonlinear equations of mathematical physics [6].

Although the schemes of the ADI methods are proved to be efficient and economic with respect to time consumption and, in most cases, unconditionally stable, they exhibit some disadvantages:

1) Their finite-difference formulations permit to consider only rectangular spatial domains (due to commutativity requirements imposed on the factorized and split operators) [7];

2) The application of the ADI schemes to the problems with Neumann and Robin boundary conditions that are varying in time encounters serious problems due to the necessity of evaluation or approximation of these boundary conditions at the intermediate steps of the scheme [8];

3) When applied to the solution of nonlinear heat equations, the operators constituting an ADI scheme do not commute, thus leading to the loss of unconditional stability of the scheme [6].

The first of the above disadvantages can be overcome either by the use of finite elements methods in conjunction with operator splitting techniques, or by domain decomposition techniques. The second and the third disadvantages present an important problem for the successful application of the ADI scheme. To the best of our knowledge, no complete stability analysis for an ADI scheme applied to the nonlinear heat equation in a three-dimensional spatial domain is available in the literature, thus motivating this work.

Another motivation for the present work is the application of ADI scheme to the modeling of heat transfer in large scale environmental systems (e.g., large areas of permafrost soils) which, in the case of purely explicit finite-differences schemes, imposes stiff constraints on the time-step in order to guarantee the numerical stability. At the same time, implementation of implicit schemes can often lead to much greater computing expenses than that of explicit schemes, especially for the problems with rapidly changing coefficients in complex geometries and substantially nonhomogeneous meshes. Thus, in modeling of heat transfer in large scale systems the necessity of making an optimal choice between explicit and implicit schemes arises. In case of finite-elements method, applied to modeling of processes in permafrost soils, the analysis of numerical stability appears to be so complex, that the stability criterion is often established empirically [9].

In the present paper we discuss the application of the ADI Douglas – Rachford scheme to the solution of Stefan problem in porous permafrost soils. The paper is organized as follows: next section contains the problem formulation and some assumptions that will be used in the proof of numerical stability of the ADI scheme while section 3 exposes the proof itself. Section 4 presents some numerical results and is followed by Conclusions.

This third Note is dedicated to the discussion of application of the 3D Douglas – Rachford ADI scheme to the solution of a non-linear heat equation. We will discuss the Newton – Raphson method and the “method of frozen coefficients”.

Before starting discussion of how to approach solution of a non-linear equation, let us define what will be called a non-linear heat equation.
Below, we will consider a heat equation with temperature dependent thermal conductivity and heat capacity. Thus, the coefficients of the heat equation appear to be temperature dependent:

This second Note is dedicated to the discussion of the case of 3-rd type boundary conditions posed on the edges of the modeling region and will present an outline of the algorithm for the account of both the Dirichlet and 3-rd type boundary conditions on different edges.

Before passing to the discussion of that scheme, a few words should be said about another ADI scheme – namely 2D Peaceman – Rachford finite difference scheme. The correct account of the intermediate boundary conditions for both Dirichlet and 3-rd order types has been thoroughly discussed in [1]. One essential difference between the 3D Douglas – Rachford and 2D Peaceman – Rachford schemes is that in the latter scheme the spatial coordinates enter symmetrically, such that the second equation (in 2D Peaceman – Rachford) contains finite difference representations of both spatial derivatives. This fact leads to a cumbersome infer of relation between the values of the unknown function on different edges – for details see [1].

Surprisingly, although the Douglas – Rachford scheme is designed for a 3D spatial region, it is easier to treat the 3-rd order boundary conditions for it correctly due to the “non-symmetrical” entrance of the spatial coordinates in three equations – see Eq. (1) – (3):